Reduction of a matrix depending on parameters to a diagonal form by addition operations

TLDR: In this paper, it was shown that any n by n matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial.

Abstract: It is shown that any n by n matrix with determinant 1 whose entries are real or complex continuous functions on a finite dimensional normal topological space can be reduced to a diagonal form by addition operations if and only if the corresponding homotopy class is trivial, provided that n 5$ 2 for real-valued functions; moreover, if this is the case, the number of operations can be bounded by a constant depending only on n and the dimension of the space. For real functions and n = 2, we describe all spaces such that every invertible matrix with trivial homotopy class can be reduced to a diagonal form by addition operations as well as all spaces such that the number of operations is bounded. Introduction. Let X be a topological space Rx the ring of all continuous functions X -* R (the reals), Rx the subring of bounded functions. For any natural number n and a ring A, MnA denotes the ring of all n by n matrices over A. A matrix a in MnRX can be regarded as a real matrix depending continuously on a parameter which ranges over X, or as a continuous map X -+ MnR. Assume now that det(a) = 1, i.e. a E SLnRX. We want to reduce a to the identity matrix ln by addition operations, i.e. represent a as a product of elementary matrices ai7, where a E A = RX, 1 < i :$ j < n. Since the subgroup EnA of SLnA generated by all elementary matrices is normal [6], it does not matter whether we use row or column addition operations, or both. Note that, by the Whitehead lemma, every diagonal matrix in SLnA is a product of 4(n -1) elementary matrices (for any commutative ring A), so a matrix a in SLnA, can be reduced to ln if and only if it can be reduced to a diagonal form. When X is a point, so A = Rx = R, it is well known that this can be done. Moreover [3, Remark 10 with sr(R) = m = 1], this can be done using at most (n 1)(3n/2 + 1) addition operations. For an arbitrary X, a homotopy obstruction may exist which prevents the reduction. Namely, the addition operations do not change the homotopy class ir(a) of the corresponding map X -+ SLnR. So if this class is not trivial, the reduction is impossible. Assume now that the homotopy class r(a) is trivial (for example, this is always the case when X is contractible). Is it possible to reduce a to ln by addition operations, i.e. does a belong to the subgroup EnRX of SLnRX generated by elementary matrices)? If yes, how many operations are needed? Received by the editors March 2, 1987 and, in revised form, June 18, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 18F25. ? 1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page